# On the chain rule and change of variables of integrals

Originally published at 狗和留美者不得入内. You can comment here or there.

Theorem 1 (Chain rule) Let , , where and are open in , such that are differentiable on their respective domains. Then is also differentiable on , with for all .

Proof: We first assume that there exists a neighborhood of for which . This happens in the case of by inverse function theorem. In that case, by the definition of derivative and its properties, we have

In the case of , we have that for all ,

From this, we easily verifies that , which means that is differentiable at and in the case of , must hold as well.

Lemma 1 Let , be differentiable with and . Then,

Instead of , one can also use any closed interval of .

Proof: Follows directly from Fundamental Theorem of Calculus. See Theorem 2 (Newton-Leibniz axiom) of [1].

Lemma 1 is a statement of invariance of integral along parameterized smooth paths with the same endpoints.

Theorem 2 (Change of variables or u-substitution in integration) Let be any differentiable function of on , which is continuous on , and be Riemann integrable on intervals in its domain. Then,

Proof: Let be an antiderivative of . By the Fundamental Theorem of Calculus, it suffices to show that the left hand side of is equal to , which can be done by applying Lemma 1 accordingly.

Theorem 3 (Integration by parts) Let be differentiable functions on and continuous on . Then,

Proof: We have

Rearranging the above completes the proof.

References

Tags:
• #### Why the 1700 Japanese could have discovered calculus and modern science by 2200 or 2700

Originally published at 狗和留美者不得入内. You can comment here or there. Some thoughts on Greek, Persian, Arab, Indian, and Chinese science In the…

• #### 知乎被留美王八蛋把控了，连我的内容为数学推导的评论都得删

没错，我不小心用了“傻逼”这个词，可是内容基本完全是数学。因为违规，又被禁了一天

• #### Selberg on Ramanujan

昨晚，我想起了大挪威数学家 Atle Selberg。他是个很有趣的人，好像在 1943 年挪威被纳粹德国占领时获得了奥斯陆大学的博士。好像他还参军了，也坐牢了。对于他的工作我知道他和 Erdos…

• Post a new comment

#### Error

Anonymous comments are disabled in this journal

default userpic